Show that the divergence of the vector 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is the limiting ratio of the surface flux to the enclosed volume.(For the surface, use the sphere with center LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ2RidGNEY3RjotRjE2JVEid0YnRjRGN0Y+Rj5GPg== and radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.)

Set coordinate systemTools\342\211\273Tasks\342\211\273Browse: Calculus - Vector\342\211\273Vector Algebra and Settings\342\211\273Set Coordinates\342\211\273The More Common Systems

The More Common Coordinate SystemsCartesian [x\134,y],Cartesian [x\134,y\134,z],Cartesian - other,polar,cylindrical,sphericaluse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
module()
local q;
Do(q = %ComboBox1):
if q=Cartesian [x,y] then Do(%MathContainer2=MathML[Export]([x,y]));
elif q=Cartesian [x,y,z] then Do(%MathContainer2=MathML[Export]([x,y,z]));
elif q=Cartesian - other then Do(%MathContainer2=NULL);
elif q=polar then Do(%MathContainer2=NULL);
elif q=cylindrical then Do(%MathContainer2=NULL);
elif q=spherical then Do(%MathContainer2=NULL);
end if:
Do(%MathContainer3 = NULL);
end module;
end use;
use DocumentTools,VectorCalculus in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
module()
local q,q1,q2,q3;
Do(q = %ComboBox1);
if q<>Cartesian [x,y] and q<> Cartesian [x,y,z] then Do(q1 = %MathContainer2);
if type([q1],'list'('list')) then q2:=q1;
elif type([q1],'list') then q2:=[q1];end if;
if q2=NULL then return;end if;
end if;
if q=Cartesian [x,y] then SetCoordinates(cartesian[x,y]);
elif q=Cartesian [x,y,z] then SetCoordinates(cartesian[x,y,z]);
elif q=Cartesian - other then if nops(q2)=3 then SetCoordinates(parse(StringTools:-Join(["cartesian",convert(q2,string)],""))) else return;end if;
elif q=polar then if nops(q2)=2 then SetCoordinates(parse(StringTools:-Join(["polar",convert(q2,string)],""))) else return;end if;
elif q=cylindrical then if nops(q2)=3 then SetCoordinates(parse(StringTools:-Join(["cylindrical",convert(q2,string)],""))) else return;end if;
elif q=spherical then if nops(q2)=3 then SetCoordinates(parse(StringTools:-Join(["spherical",convert(q2,string)],""))) else return;end if;
end if;
q3 := GetCoordinates();
Do(%MathContainer3 = MathML[Export]([op(0,q3),[op(q3)]]));
end module;
end use;

The field FUsing angle brackets, enter the field F as a vector.Context Menu: Assign to a Name\342\211\273FJSFHDivergence by differential operatorCommon Symbols or Operators palette:
Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictRiw2LVEpJm1pZGRvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGSy1JI21pR0YkNiVRIkZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLw==Context Menu: Evaluate and Display InlineLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Flux through Surface of SphereLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Compute surface flux, that is, 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the surface of a sphere of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YyTools\342\211\273Tasks\342\211\273Browse: Calculus - Vector\342\211\273Integration\342\211\273Flux\342\211\2733-D\342\211\273Through a Sphere

Flux through a SphereLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Ratio of Flux to VolumeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Divide by the volume of a sphere, etc.Copy/paste the flux and press the Enter key.Context Menu: FactorDivide by 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

JSFHThe field FUsing angle brackets, enter the field F as a vector.Context Menu: Assign to a Name\342\211\273FLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Divergence by differential operatorCommon Symbols or Operators palette:
Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictRiw2LVEpJm1pZGRvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGSy1JI21pR0YkNiVRIkZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLw==Context Menu: Evaluate and Display InlineJSFH

Flux through Surface of SphereJSFH

Compute surface flux, that is, 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the surface of a sphere of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YyTools\342\211\273Tasks\342\211\273Browse: Calculus - Vector\342\211\273Integration\342\211\273Flux\342\211\2733-D\342\211\273Through a Sphere

Divide by the volume of a sphere, etc.Copy/paste the flux and press the Enter key.Context Menu: FactorDivide by 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